Finite Group DatabasebetaFinite Group Databasebeta| order | 35322 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊D841 |
| unambiguous semidirect product | C841⋊AGL(1,𝔽7) |
| unambiguous semidirect product | D5887⋊C3 |
| Fitting, derived, Hall | C7×C841 ⋊ C2×C3 |
| Hall | ΓL(1,𝔽8)×C841 ⋊ C2 |
| Hall | D5887 ⋊ C3 |
| Sylow | C841 ⋊ AGL(1,𝔽7) |
| Hall | ΓL(1,𝔽8) ⋊ D841 |
| Sylow | C7 ⋊ C3×D841 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C7×C841 | C2×C3 |
| Frattini | C29 | ΓL(1,𝔽8)⋊D29 |
| socle | C7×C29 | C3×D29 |
| LCS | G ▷ C7×C841 | |
| UCS | C1 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 848 |
| transitive degree | μt(G) = 5887 |
A minimal faithful permutation representation — 2 generators on 848 points:
(2,6,5,3,4,7)(9,11)(10,14)(12,17)(13,20)(15,23)(16,22)(18,27)(19,30)(21,32)(24,36)(25,35)(26,34)(28,41)(29,44)(31,46)(33,48)(37,53)(38,52)(39,51)(40,50)(42,59)(43,62)(45,64)(47,66)(49,68)(54,74)(55,73)(56,72)(57,71)(58,70)(60,81)(61,84)(63,86)(65,88)(67,90)(69,92)(75,99)(76,98)(77,97)(78,96)(79,95)(80,94)(82,107)(83,110)(85,112)(87,114)(89,116)(91,118)(93,120)(100,128)(101,127)(102,126)(103,125)(104,124)(105,123)(106,122)(108,137)(109,140)(111,142)(113,144)(115,146)(117,148)(119,150)(121,152)(129,161)(130,160)(131,159)(132,158)(133,157)(134,156)(135,155)(136,154)(138,171)(139,174)(141,176)(143,178)(145,180)(147,182)(149,184)(151,186)(153,188)(162,198)(163,197)(164,196)(165,195)(166,194)(167,193)(168,192)(169,191)(170,190)(172,209)(173,212)(175,214)(177,216)(179,218)(181,220)(183,222)(185,224)(187,226)(189,228)(199,239)(200,238)(201,237)(202,236)(203,235)(204,234)(205,233)(206,232)(207,231)(208,230)(210,251)(211,254)(213,256)(215,258)(217,260)(219,262)(221,264)(223,266)(225,268)(227,270)(229,272)(240,284)(241,283)(242,282)(243,281)(244,280)(245,279)(246,278)(247,277)(248,276)(249,275)(250,274)(252,297)(253,300)(255,302)(257,304)(259,306)(261,308)(263,310)(265,312)(267,314)(269,316)(271,318)(273,320)(285,333)(286,332)(287,331)(288,330)(289,329)(290,328)(291,327)(292,326)(293,325)(294,324)(295,323)(296,322)(298,347)(299,350)(301,352)(303,354)(305,356)(307,358)(309,360)(311,362)(313,364)(315,366)(317,368)(319,370)(321,372)(334,386)(335,385)(336,384)(337,383)(338,382)(339,381)(340,380)(341,379)(342,378)(343,377)(344,376)(345,375)(346,374)(348,401)(349,404)(351,406)(353,408)(355,410)(357,412)(359,414)(361,416)(363,418)(365,420)(367,422)(369,424)(371,426)(373,428)(387,443)(388,442)(389,441)(390,440)(391,439)(392,438)(393,437)(394,436)(395,435)(396,434)(397,433)(398,432)(399,431)(400,430)(402,403)(405,459)(407,461)(409,463)(411,465)(413,467)(415,469)(417,471)(419,473)(421,475)(423,477)(425,479)(427,481)(429,483)(444,484)(445,497)(446,496)(447,495)(448,494)(449,493)(450,492)(451,491)(452,490)(453,489)(454,488)(455,487)(456,486)(457,485)(458,511)(460,513)(462,515)(464,517)(466,519)(468,521)(470,523)(472,525)(474,527)(476,529)(478,531)(480,533)(482,535)(498,536)(499,548)(500,547)(501,546)(502,545)(503,544)(504,543)(505,542)(506,541)(507,540)(508,539)(509,538)(510,537)(512,561)(514,563)(516,565)(518,567)(520,569)(522,571)(524,573)(526,575)(528,577)(530,579)(532,581)(534,583)(549,584)(550,595)(551,594)(552,593)(553,592)(554,591)(555,590)(556,589)(557,588)(558,587)(559,586)(560,585)(562,607)(564,609)(566,611)(568,613)(570,615)(572,617)(574,619)(576,621)(578,623)(580,625)(582,627)(596,628)(597,638)(598,637)(599,636)(600,635)(601,634)(602,633)(603,632)(604,631)(605,630)(606,629)(608,649)(610,651)(612,653)(614,655)(616,657)(618,659)(620,661)(622,663)(624,665)(626,667)(639,668)(640,677)(641,676)(642,675)(643,674)(644,673)(645,672)(646,671)(647,670)(648,669)(650,687)(652,689)(654,691)(656,693)(658,695)(660,697)(662,699)(664,701)(666,703)(678,704)(679,712)(680,711)(681,710)(682,709)(683,708)(684,707)(685,706)(686,705)(688,721)(690,723)(692,725)(694,727)(696,729)(698,731)(700,733)(702,735)(713,736)(714,743)(715,742)(716,741)(717,740)(718,739)(719,738)(720,737)(722,751)(724,753)(726,755)(728,757)(730,759)(732,761)(734,763)(744,764)(745,770)(746,769)(747,768)(748,767)(749,766)(750,765)(752,777)(754,779)(756,781)(758,783)(760,785)(762,787)(771,788)(772,793)(773,792)(774,791)(775,790)(776,789)(778,799)(780,801)(782,803)(784,805)(786,807)(794,808)(795,812)(796,811)(797,810)(798,809)(800,817)(802,819)(804,821)(806,823)(813,824)(814,827)(815,826)(816,825)(818,831)(820,833)(822,835)(828,836)(829,838)(830,837)(832,841)(834,843)(839,844)(840,845)(842,847)(846,848);(1,2,5,6,4,7,3)(8,160,536,225,43,279,677,472,175,434,769,694,407,586,825,798,509,515,757,716,343,258,575,598,204,86,316,445,101,14,197,482,185,28,328,704,417,139,490,793,656,351,630,837,776,456,565,783,682,294,306,621,552,167,114,368,389,76,23,238,427,149,70,381,666,361,108,542,808,614,299,670,845,750,400,611,805,644,249,358,663,502,134,146,424,336,55,36,283,371,117,95,438,624,309,190,590,786,568,252,706,848,720,347,653,823,602,208,414,701,449,105,182,479,287,38,53,332,319,89,124,494,578,261,231,634,760,518,374,738,842,686,404,691,813,556,171,469,735,393,80,222,533,242,25,74,385,271,65,157,546,528,217,276,674,730,464,431,766,832,648,459,725,795,506,212,523,713,340,59,266,583,201,16,99,442,227,45,194,594,474,177,325,710,696,409,487,790,818,606,513,755,773,453,256,573,679,291,84,314,549,164,11,128,484,187,29,235,638,419,141,378,742,658,353,539,810,800,560,563,781,747,397,304,619,641,246,112,366,499,131,20,161,429,151,18,280,668,363,109,435,770,616,301,587,826,778,510,609,803,717,344,356,661,599,205,144,422,446,102,32,198,373,119,50,329,626,311,82,491,788,570,253,631,838,752,457,651,821,683,295,412,699,553,168,180,477,390,77,48,239,321,91,71,382,580,263,154,543,762,520,210,671,844,722,401,689,835,645,250,467,733,503,135,220,531,337,56,68,284,273,67,96,439,530,219,191,591,732,466,322,707,834,688,402,723,828,603,209,521,763,450,106,264,581,288,39,92,333,229,47,125,495,476,179,232,635,698,411,375,739,820,650,511,753,814,557,254,571,744,394,81,312,627,243,26,120,386,189,31,158,547,421,143,277,675,660,355,432,767,802,608,561,779,796,507,302,617,714,341,110,364,596,202,17,152,443,153,19,195,595,365,111,326,711,618,303,488,791,780,562,607,801,774,454,354,659,680,292,142,420,550,165,30,188,387,121,12,236,628,313,83,379,743,572,255,540,811,754,512,649,819,748,398,410,697,642,247,178,475,500,132,46,228,334,93,34,281,582,265,60,436,764,522,211,588,827,724,458,687,833,718,345,465,731,600,206,218,529,447,103,66,272,285,69,51,330,532,221,122,492,734,468,172,632,836,690,403,721,843,684,296,519,761,554,169,262,579,391,78,90,320,240,49,72,383,478,181,155,544,700,413,274,672,822,652,348,751,839,646,251,569,787,504,136,310,625,338,57,118,372,199,33,97,440,423,145,192,592,662,357,323,708,804,610,485,777,829,604,300,615,771,451,107,362,667,289,40,150,428,162,21,126,496,367,113,233,636,620,305,376,740,782,564,537,799,815,558,352,657,745,395,140,418,639,244,27,186,483,129,13,159,548,315,85,278,676,574,257,433,768,756,514,585,817,797,508,408,695,715,342,176,473,597,203,44,226,444,100,9,196,584,267,61,327,712,524,213,489,792,726,460,629,831,775,455,463,729,681,293,216,527,551,166,64,270,388,75,22,237,534,223,42,380,736,470,173,541,812,692,405,669,841,749,399,517,759,643,248,260,577,501,133,88,318,335,54,35,282,480,183,94,437,702,415,138,589,824,654,349,705,847,719,346,567,785,601,207,308,623,448,104,116,370,286,37,52,331,425,147,123,493,664,359,230,633,806,612,298,737,846,685,297,613,807,555,170,360,665,392,79,148,426,241,24,73,384,369,115,156,545,622,307,275,673,784,566,430,765,840,647,350,655,794,505,137,416,703,339,58,184,481,200,15,98,441,317,87,193,593,576,259,324,709,758,516,486,789,830,605,406,693,772,452,174,471,678,290,41,224,535,163,10,127,497,269,63,234,637,526,215,377,741,728,462,538,809,816,559,461,727,746,396,214,525,640,245,62,268,498,130)Polycyclic presentation:
2,3,7,29^2|d^29=e^28,c^a=c^6,d^a=d^28e,e^a=e^28,c^b=c^42107 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(35322, 8);